The value of `((a^(x))/(a^(y)))^(x+y) xx ((a^(y))/(a^(z)))^(y+z) xx ((a^(z))/(a^(x)))^(z+x)` is

Options

A. 0
B. `(1)/(y)`
C. 1
D. `(1)/(xyz)`

Answer

To solve the problem, we need to evaluate the expression:

\[ \left( \frac{a^x}{a^y} \right)^{x+y} \times \left( \frac{a^y}{a^z} \right)^{y+z} \times \left( \frac{a^z}{a^x} \right)^{z+x} \]

### Step 1: Simplify Each Term
We can simplify each term in the expression using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \).

1. First term:
\[ \left( \frac{a^x}{a^y} \right)^{x+y} = \left( a^{x-y} \right)^{x+y} = a^{(x-y)(x+y)} \]

2. Second term:
\[ \left( \frac{a^y}{a^z} \right)^{y+z} = \left( a^{y-z} \right)^{y+z} = a^{(y-z)(y+z)} \]

3. Third term:
\[ \left( \frac{a^z}{a^x} \right)^{z+x} = \left( a^{z-x} \right)^{z+x} = a^{(z-x)(z+x)} \]

### Step 2: Combine the Terms
Now we can combine all three terms:
\[ a^{(x-y)(x+y)} \times a^{(y-z)(y+z)} \times a^{(z-x)(z+x)} = a^{(x-y)(x+y) + (y-z)(y+z) + (z-x)(z+x)} \]

### Step 3: Expand Each Term
Now we will expand each of the products in the exponent:

1. For \( (x-y)(x+y) \):
\[ (x-y)(x+y) = x^2 - y^2 \]

2. For \( (y-z)(y+z) \):
\[ (y-z)(y+z) = y^2 - z^2 \]

3. For \( (z-x)(z+x) \):
\[ (z-x)(z+x) = z^2 - x^2 \]

### Step 4: Combine the Expanded Terms
Now we can combine these results:
\[ x^2 - y^2 + y^2 - z^2 + z^2 - x^2 \]

### Step 5: Simplify the Expression
Notice that all terms cancel out:
\[ x^2 - y^2 + y^2 - z^2 + z^2 - x^2 = 0 \]

### Step 6: Final Result
Thus, we have:
\[ a^0 = 1 \]

### Conclusion
The value of the given expression is:
\[ \boxed{1} \]